Field

Bonding models

Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.

1s with 1s at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 1s at 50% of its density, |ψ| = 1.48e-1.

Overlap decides

Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it.

1s with 2px at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 2px at 50% of its density, |ψ| = 3.16e-2.

Exactly zero

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

sp3 hybrids. The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.

Hybrids are a basis

An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

Hückel theory and what it gets right

A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

Aromaticity as a computed shell closure

Hückel's 4n+2 rule is not a rule. It is what comes out when every ring size is filled and asked whether its highest occupied shell came out full — and the calculation that answers does not know the phrase.

π bond orders in naphthalene. Each bond drawn with a thickness set by its computed pi bond order, and the number printed beside it. The orders come from the eigenvectors and cost nothing extra once the matrix is diagonalised.

Bond order from the eigenvectors

Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The localisation transformation, demonstrated

Four equivalent bonds or one a₁ and three t₂ — two descriptions of methane's bonding electrons that disagree about everything except the electron density, which they agree about to the last bit a double can hold.

Two orbitals, 4 electrons, S = 0 and S = 0.25. Two interacting orbitals with 4 electrons in them, drawn twice: once with the overlap set to zero and once with it kept at 0.25. Dropping the overlap makes the two shifts equal, which is the picture usually taught; keeping it makes the upper level rise by more than the lower falls, which is why four electrons in two orbitals is a repulsion.

The antibonding level goes up more

The two-level diagram every course draws is symmetric, and the symmetry is an artefact of setting the overlap to zero. Keep it, and the upper level rises further than the lower one falls — which is why helium has no molecule and why closed shells push each other apart.

pyridine against benzene. The Hückel levels of pyridine beside those of benzene, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

Hückel with a heteroatom

A hydrocarbon's Hückel matrix contains no fitted number at all. Put one nitrogen in the ring and two arrive at once, the levels stop being symmetric about α, and the delocalisation energy stops being quotable — all from changing three entries in a matrix.

water: 4 valence bands. The measured valence photoelectron bands of water, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

Water's lone pairs are not a pair

Every course draws two equivalent lone pairs on water, pointing away from the hydrogens like a pair of ears. Its photoelectron spectrum shows the two bands they would produce at 12.6 and 14.7 electronvolts, two point one apart, in different symmetry species.

Two sites, two electrons, every level exactly. The four states of a two-site Hubbard model against the on-site repulsion, in units of the hopping. The left-hand edge is the one-electron answer — -2t — and the dashed line is the lowest state with the spins parallel, which cannot hop at all and so does not move with U.

The smallest many-electron calculation

A Hückel energy comes from a model with one electron in it. Add a single term — a cost for two electrons on the same site — and the problem stops being a matrix of size n and becomes a matrix over configurations. Four sites give thirty-six of them, which is small enough to solve exactly, and the answers correct two things the one-electron model got wrong.

How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 2, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is what the state actually pays in repulsion, which peaks and then falls because the electrons have stopped meeting.

Where molecular orbital theory dissociates

The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.

The same ring, three charges

Three carbons in a ring are aromatic with two π electrons, a doublet with three, and a triplet with a delocalisation energy of exactly zero with four. Nothing about the molecule changed but the count, and adding electrons to a π system can make its π binding energy fall.

The valence-bond and molecular-orbital directions in one plane. The valence-bond function and the molecular-orbital function as two directions, at the angle their overlap requires — 45.0°, since they overlap by 0.7071. The exact ground state lies in the plane they span at every repulsion, to twelve decimal places, and swings from one to the other as the repulsion grows without ever arriving. Neither picture is a special case of the other and the answer is not either of them.

Two pictures, one plane

Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.

The same dimer, one, twice and three times over. Independent Hubbard dimers with nothing between them, solved exactly and solved in a space with the configurations that make more than one of them ionic thrown away. The exact energy is exactly additive; the truncated one is exact for a single dimer, because there is nothing there to throw away, and falls behind by 0.193 for two and 0.485 for three. The error per dimer grows, which is what makes a method size-inconsistent rather than merely approximate.

A method that is not additive

Put a molecule next to a copy of itself, far enough away that they do not interact, and the exact energy doubles exactly. A truncated calculation does not — because the truncation forbids both halves being excited at once, which is something the pair can do and neither half can.

Overlap does not always fall as the atoms are pulled apart. The overlap integral of two pairs of orbitals against the separation between their centres, with the turning point and the sign change of each found by bisection on the integral itself. A pair with a radial node in it does not fall monotonically and does not keep one sign.

Closer is not more overlap

Two 1s orbitals overlap more the closer they are, and every curve drawn from that pair says the same thing. Put a radial node into one of them and the rule fails: a 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr, and less again beyond — and two 2p orbitals head-on change sign at 5.06 bohr and are more strongly coupled at eight bohr than at four.

One double bond, two descriptions. A carbon–carbon double bond drawn twice in the plane perpendicular to the molecule: as a σ orbital along the axis with a π orbital above and below it, and as two equivalent bent bonds tilted 50.8 degrees either side of the axis. The two descriptions are related by a rotation and have the same density everywhere.

Two bent bonds, or a σ and a π

A carbon–carbon double bond is drawn two ways and the two look like rival claims about what is there. They are one occupied space in two bases, related by a rotation of exactly forty-five degrees: the density is identical to the last bit a double holds, the bent pair are sp⁵ hybrids at 50.77° to the axis, and their charge sits 0.235 Å off the plane of the molecule where neither canonical orbital's does.

The same difference, at five repulsions. The charge transferred to the more electronegative of two atoms against the difference in their orbital energies, at five strengths of the repulsion between the two electrons. Only the topmost curve is the two-level result; every other one moves far less charge at the same difference.

A difference does not make a transfer

Every electronegativity scale reports one thing: how far apart two atoms are in their appetite for electrons. Put that difference into a model with repulsion in it and the charge it actually moves is not determined at all — the same difference of one moves 0.447 of an electron with no repulsion and 0.0019 with a strong one, a factor of two hundred and forty-two, and nothing on any scale distinguishes the two cases.

The ionic weight against how much ionic structure is in the wavefunction. The percentage each convention calls ionic, at a fixed structure overlap, as the amount of ionic structure in the wavefunction is raised from none to the molecular orbital value. They meet at both ends of the sweep and disagree everywhere between. Every curve is a weight and every set sums to one.

A weight that depends on how it is weighed

The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

Two derivatives of the same energy, and only one is tabulated. The 18 elements of the electronegativity tables, placed by their chemical potential — half the sum of the ionisation energy and the electron affinity, which is the Mulliken electronegativity — against their hardness, half the difference of the same two numbers. Hardness runs from 1.92 to 7.3 electronvolts, a factor of 3.8, and does not follow the horizontal axis. Ringed points are the three elements whose anion is not bound, so whose affinity is not a measurement.

The quantity no scale prints

Every electronegativity table is half of a calculation. The other half is the hardness — half the difference of the same two measurements the Mulliken scale is half the sum of — and it varies by a factor of 3.8 across eighteen elements. Put both halves in and B–F, with an electronegativity difference of 6.12 electronvolts, moves less charge than lithium iodide, whose difference is 3.75.

seven sets of parameters, one spectrum. Benzene's π levels from seven sets of the three Hückel parameters, each fitted to reproduce the two measured ionisation energies exactly. The two occupied levels sit at the same energy in every column, because that is what was fitted. The empty level moves from -3.15 to 4.69 electronvolts across the family, and the resonance integral from -3.05 to -7.66.

One spectrum, a line of models

Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.

Which numbers survive a change of frame, and which are coordinates. Every quantity this field quotes, against the four things that can be changed without changing the molecule: the zero of energy, the unit, the reference state a stabilisation is measured from, and what a "per" quantity is divided by. A filled mark is a quantity that moves. three of the 9 survive all four, and every one of them is a property of the eigenvectors rather than of the energies. The first two changes are exact symmetries of the model, so a quantity that moves under either is a coordinate and not a quantity at all.

Which numbers carry a frame

A Hückel calculation is written in two numbers nobody computes and quoted against reference states nobody measures, so every quantity it prints is a quantity in a frame. Nine of them, tested against four changes of frame: three survive all four, and the one that survives both exact symmetries and looks safest — a dimensionless ratio between two molecules — is the most fragile of all, because against one reference its denominator is exactly zero.

One pi energy, two delocalisation energies. For each of six rings: the computed pi energy, the energy of the same atoms with one bond deleted, and the delocalisation energy that follows from each of the two reference states. Every entry is an eigenvalue sum; the two right-hand columns differ only in what was subtracted from the second column.

The frame that was allowed to relax

Four changes of frame were tested on nine quantities and none of them could move a bond order, because a bond order is a property of the eigenvectors and every change left the eigenvectors alone. Letting the geometry answer back does move them — butadiene's central bond falls from 0.4472 to 0.3676 — and it moves naphthalene's the other way, because its weakest bond is the one two rings share and relaxation strengthens it.

Carbon is not one number. The charge equalisation puts on carbon in the four fluoromethanes, and on hydrogen in the three that have one. Carbon's runs from 0.07 to 0.31, and hydrogen's changes sign between methane and fluoromethane, so the same C–H bond is polarised one way in one molecule and the other way in the next. A table with one number for carbon is printing the value an iteration starts from.

The value that only exists in the bond

A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.

A parameter that never finds a value

Show a two-parameter model six measurements instead of two and it stops being underdetermined and starts being wrong. Adding the third parameter improves the fit by one part in eighty, moves the resonance integral by a factor of four, and never finds a best value at all — because nine tenths of the error is a term the model does not have.

The hybrids do not point at the atoms. For each cycloalkane, the angle between its two ring hybrids — fixed by orthogonality once the measured H–C–H angle has said how much s character the hydrogens take — against the angle between its carbons. Cyclopropane's differ by 45.5°, so each hybrid points 22.75° outside the bond it makes; cyclohexane's agree to 0.035°, which is the control.

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

The shared bond's response falls and reverses, at every coupling. How far the relaxation moves the most interior cross bond of an acene, against the number of rings, at three couplings. Every series falls monotonically and every one changes sign — at λ = 0.2 between 3 and 4 rings, at λ = 0.4 between 3 and 4 rings, at λ = 0.6 between 4 and 5 rings. Where it crosses is a property of the coupling; that it crosses is not. The natural reading — that the effect grows with the number of rings feeding a bond — is refused by every one of these curves.

An anomaly that is not the first of a series

Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.

Two systems a model cannot tell apart

Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.

The angle that does not have to be searched for

A carbonyl's two bent components have no symmetry making them equivalent, so the mixing that best localises them looks like something to search for and their s characters look like two different numbers. Neither happens. The localisation functional is a quadratic with no linear term, so its maximum is at forty-five degrees or at the ends — bent bonds or canonical ones, decided by one inequality, with nothing in between.

The pair that is not a tie

An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.

The filled orbital at 3 bohr, with a node in the middle of the bond. The combination the two-level model puts lower at 3 bohr, drawn as contours of the wavefunction with the two signs in the two colours. The overlap here is 0.4825, so the interaction is of the sign that fills the plus combination. There is a nodal plane through the midpoint: the pair the model calls bonded has no density at all between its nuclei.

A bond with nothing in the middle

Two head-on 2p functions have an overlap that changes sign at 5.03 bohr, and the picture of what that means is worth drawing. Below that separation the combination the model fills has a nodal plane through the midpoint of the bond, so two electrons in it put exactly nothing between the nuclei; above it the same model fills the other one. Which picture a bonding orbital has is decided by a separation.

The inequality, decided twice. The quantity the localisation criterion compares — the centroid separation over twice the off-axis dipole — computed with hydrogenic radial functions and with Slater ones at the same effective charges. Below one the localised description is a pair of bent components; above it, σ and π. The hydrogenic answer is 1.8090 and the Slater answer is 0.3937, and they are on opposite sides.

The node that decided a picture

A hydrogenic calculation cannot say whether formaldehyde's localised description is two bent components or σ and π, because the integral that decides it came out at 0.00718 where it should be the largest in the molecule. Replacing the hydrogenic radial functions with nodeless Slater ones moves that overlap by a factor of ninety-seven, leaves the π overlap identical to the last bit, and flips the answer: the description is bent.

One heteroatom, and how far the relaxation carries it. The change every ring's bonds undergo when one carbon of the end ring is given a site energy, on a chain of 12 fused hexagons with its gap held open. The upper curve measures each molecule's couplings from its own mean bond order, which is what this collection's relaxation has always done; it stops falling at 4.48e-5 and stays there. The lower curve measures both from the same mean and keeps falling to 7.89e-9. Nothing about the molecule differs between them.

The floor was in the bookkeeping

A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

What a rotation group buys over a single angle. The Boys functional at the canonical orbitals, at the best mixing of the two bonds with the lone pair left alone, and at the maximum over the whole three-dimensional rotation group. The first step is 11.70 per cent and the second, which no two-orbital treatment can reach, is a further 2.50.

An interior maximum a third orbital allows

Two orbitals related by a mirror plane give a Boys functional with no linear term, so it has no interior maximum and no angle is ever searched for. Add the oxygen lone pair and the search over a three-dimensional rotation group finds one — eight and a half degrees of lone pair mixed into two bent bonds, worth a further two and a half per cent that no pair of orbitals could reach.

The reach is the molecule's

Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.

$What it would cost to destroy each refutation. For every discordant pair in the collection, the measurement error that would be needed to reverse it, as a fraction of that predictor's own spread of measurements. Nothing is quoted: the question is not what the uncertainties are but how large they would have to be. The cheapest to destroy needs 3.4 per cent of the range and the dearest needs 51.6.$

The error bar that would be needed

Two things a pair of measurements can say without any model both treat their numbers as exact. The quoted measurements carry no uncertainties, and inventing some would be worse than having none — so the question is asked the other way round. Not what the errors are, but how large they would have to be. The three refutations that seemed most worrying turn out to be the sturdiest of the lot.

$The criterion is the reciprocal of the p amplitude. The two lone pairs' mixing criterion — twice the dipole between them over the separation of their centroids — against the p fraction of the outward hybrid. The curve is 1/√(p fraction), written down rather than fitted; the marks are the numerical integrals. They agree to 15 parts in a million at every hybrid, and the criterion never falls to one, so the mixed description wins at every s character there is.$

The five figures were an identity

Two lone pairs satisfy the two-orbital mixing criterion by a margin of 1.732128, against √3 = 1.732051 — agreement to five figures on a number assembled from three integrals over a numerical grid. It is exact. The criterion is the reciprocal of the p amplitude of the outward hybrid, and everything else in the molecule cancels out of it.

The bracket has no upper end. Every price is multiplied by √2 over √(1 + r² − 2ρr), where ρ is the correlation between the two measurements' errors and r is the ratio of their sizes. For equal precision the factor is 1/√(1 − ρ), which is one at independence and unbounded at perfect correlation. The other extreme the question asked for is not a number.

The other end of the bracket is not a number

Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.

One kink, and the heteroatom's own ring notices only at the kink. The response on the heteroatom's own ring, on a straight chain of twelve and on one with a kink — two adjacent angular fusions — in the middle, as the heteroatom is moved along. The two agree to within one and a half per cent except on the kink's two rings: on the first the bent chain reads 18.9 per cent lower and on the second 3.1 per cent lower. An end halves the amplitude.

A bend is not an end

Moving a heteroatom from the end of a straight chain of twelve rings to its middle leaves the decay unchanged and halves the amplitude, which raises the question of whether a bend in the chain is a boundary of the same kind. It is not a boundary of the same kind: the kink — two adjacent angular fusions — leaves the fitted decay length within two per cent and costs the heteroatom's own ring a fifth only on the kink's first angular ring. It does multiply down the response on the rings beyond it, which a decay length does not see.

The ranking is fragile and the headline is not. The correlation required to disturb two different things. Reordering the easiest neighbouring pair needs ρ = 0.154, which is weak enough to expect. Displacing the collection's most fragile claim outright needs ρ = 0.954, which is near-perfect correlation. So a per-predictor structure scrambles the middle of the ordering and leaves the top of it alone.

The ranking moved and the headline did not

One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.

Five pairs, one boundary, and every case anybody asked about on one side. The mixing criterion's margin for every pair of orbitals available from the carbonyl and from ethene. Above one the mixed description wins. Three pairs were put to the criterion and all three clear the line; two more come from the same three orbitals and the same dipole matrices, and both fail it. The instrument discriminates — the selection did not.

The criterion that has never said no

The two-orbital mixing criterion is a tautology on a carbonyl's lone pairs, and it is tempting to contrast that with the double bond, where the criterion seems to discriminate. Both halves are wrong. The double bond's margin is wider, ethene's is infinite because symmetry puts both centroids at the same point, and the criterion has never returned a negative on any case it is usually put to — though the same three orbitals supply two pairs it refuses outright.

The A–C overlap changes by a fifth and the exact level does not move. The three levels of the trio as the overlap between A and C is raised from 0.25 by up to 0.2, with B's overlap to C held and both site energies at -13.6 eV. The two outer orbitals stop being equivalent at the first step. The lowest level falls by 0.80 eV and the highest rises by 5.24, and the middle one stays at -13.6 eV — its largest departure over the whole sweep is 2.7×10⁻¹⁴ eV, which is rounding.

A level no symmetry was protecting

A three-orbital trio keeps one level at the free-atom energy exactly, and the reason given was that its two outer orbitals are equivalent. Make them inequivalent by changing one overlap rather than one energy and the level does not move at all — not to first order, not to any order, at any energy of the third orbital. It was never the symmetry. It is allyl's non-bonding orbital, held by a count.

Three chains and ninety orderings ruled out

Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

$Two prices, and which is cheaper is a property of the claim. For each slope floor, what it costs to halve it two ways. The measurement price is a standard error on the measured rise, as a fraction of the measured range — the currency every price in this argument has been quoted in. The model price is the change in the computed run, as a fraction of the predictor's own range. They are different currencies and their ordering differs between claims: the angle strain's floor is three times cheaper to move through its model, and the spin-only moment's cannot be moved through its model at any price at all.$

A denominator needs three currencies

Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.

A floor on models written in one scale

A tie asks whether two predicted values are equal and a discordance asks whether two differences have the same sign. Both are questions about order, and a strictly increasing change of scale preserves order — so both are exactly invariant under thirteen reparameterisations of all three predictors. The slope floor asks for a ratio of differences, and it moves by factors of nineteen, ten and eleven thousand.

The unpaired electron's level stays put while its spin moves to the far end. Above, the trio's three levels as the overlap of A with C is raised from 0.25 to 0.45 while B's stays at 0.25. The lowest falls from -16.26 to -17.07 eV and the highest rises from -8.02 to -2.78 eV; the middle one, which holds the radical's unpaired electron, stays at −13.6 eV. Below, the spin on A and on B over the same change: from a half each to 0.236 on A and 0.764 on B, with the ratio of squared overlaps drawn as open circles on top.

The spin the count does not hold

Three orbitals in a row keep one level at the free-atom energy however the overlap of one end is changed, and at three electrons that level holds the radical's unpaired electron. Its energy does not move. Its spin does: from half on each end to 0.236 and 0.764 as one overlap goes from 0.25 to 0.45, exactly the squared ratio of the two overlaps, at every energy of the middle orbital. A coupling between the ends moves the level and cannot move the spin. The energy and the spin are answering to different things.

The one-bend family was one angular ring at its ends and two everywhere else. Filled circles: the worst spread within a separation class for the eight nine-ring chains built by bending one fusion, placed at the angular rings each actually has. The chains bent at the first and last fusions have one angular ring, next to an end; the six bent in between have two adjacent angular rings whose turns cancel. The lower line is a single angular ring at each interior position, running 3.05, 3.66, 3.46, 3.03, 3.46, 3.66, 3.05; the upper line is two adjacent ones, running 4.30, 4.60, 4.06, 4.06, 4.60, 4.30. Every published point lies on one line or the other.

The scatter counts angular rings

Eight nine-ring chains built by turning one fusion looked like one bend moved along a molecule. They were two families: one angular ring when the turned fusion is at an end, two adjacent angular rings everywhere else. Built from stated angular rings instead, twenty-seven chains show the ring-current scatter ignores which way a ring turns, does not grow with how many turn, and is mostly explained by one count per pair — each angular ring between two rings multiplies their response by 0.60, each one under them by 0.65.

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